channels.reduction

The reduction channel.

Module Contents

Functions

reduction(dim[, k])

Produce the reduction map or reduction channel [1].

channels.reduction.reduction(dim, k=1)

Produce the reduction map or reduction channel [1].

If k = 1, this returns the Choi matrix of the reduction map which is a positive map on dim-by-dim matrices. For a different value of k, this yields the Choi matrix of the map defined by:

\[R(X) = k * \text{Tr}(X) * \mathbb{I} - X,\]

where \(\mathbb{I}\) is the identity matrix. This map is \(k\)-positive.

Examples

Using toqito, we can generate the \(3\)-dimensional (or standard) reduction map as follows.

>>> from toqito.channels import reduction
>>> reduction(3)
array([[ 0.,  0.,  0.,  0., -1.,  0.,  0.,  0., -1.],
       [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
       [-1.,  0.,  0.,  0.,  0.,  0.,  0.,  0., -1.],
       [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.],
       [-1.,  0.,  0.,  0., -1.,  0.,  0.,  0.,  0.]])

References

[1] (1,2)

Wikipedia. Reduction criterion. https://en.wikipedia.org/wiki/Reduction_criterion.

Parameters:

• dim (int) – A positive integer (the dimension of the reduction map).

• k (int) – If this positive integer is provided, the script will instead return the Choi matrix of the following linear map: Phi(X) := K * Tr(X)I - X.

Returns:

The reduction map.

Return type:

numpy.ndarray